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Symmetry in crystals 
CARMELO GIACOVAZZO 
The crystalline state and isometric operations 
Matter is usually classified into three states: gaseous, liquid, and solid. 
Gases are composed of almost isolated particles, except for occasional 
collisions; they tend to occupy all the available volume, which is subject to 
variation following changes in pressure. In liquids the attraction between 
nearest-neighbour particles is high enough to keep the particles almost in 
contact. As a consequence liquids can only be slightly compressed. The 
thermal motion has sufficient energy to move the molecules away from the 
attractive field of their neighbours; the particles are not linked together 
permanently, thus allowing liquids to flow. 
If we reduce the thermal motion of a liquid, the links between molecules 
will become more stable. The molecules will then cluster together to form 
what is macroscopically observed as a rigid body. They can assume a 
random disposition, but an ordered pattern is more likely because it 
corresponds to a lower energy state. This ordered disposition of molecules is 
called the crystalline state. As a consequence of our increased understand- 
ing of the structure of matter, it has become more convenient to classify 
matter into the three states: gaseous, liquid, and crystalline. 
Can we then conclude that all solid materials are crystalline? For 
instance, can common glass and calcite (calcium carbonate present in 
nature) both be considered as crystalline? Even though both materials have 
high hardness and are transparent to light, glass, but not calcite, breaks in a 
completely irregular way. This is due to the fact that glass is formed by long, 
randomly disposed macromolecules of silicon dioxide. When it is formed 
from the molten state (glass does not possess a definite melting point, but 
becomes progressively less fluid) the thermal energy which remains as the 
material is cooled does not allow the polymers to assume a regular pattern. 
This disordered disposition, characteristic of the liquid state, is therefore 
retained when the cooling is completed. Usually glasses are referred to as 
overcooled liquids, while non-fluid materials with a very high degree of 
disorder are known as amorphous solids. 
A distinctive property of the crystalline state is a regular repetition in the 
three-dimensional space of an object (as postulated as early as the end of 
the eighteenth century by R. J. Haiiy), made of molecules or groups of 
molecules, extending over a distance corresponding to thousands of 
molecular dimensions. However, a crystal necessarily has a number of 
defects at non-zero temperature and/or may contain impurities without 
losing its order. Furthermore: 
1. Some crystals do not show three-dimensional periodicity because the 
2 1 Carmelo Giacovazzo 
basic crystal periodicity is modulated by periodic distortions incom- 
mensurated with the basic periods (i.e. in incommensurately modulated 
structures, IMS). It has, however, been shown (p. 171 and Appendix 
3.E) that IMSs are periodic in a suitable (3 + d)-dimensional space. 
2. Some polymers only show a bi-dimensional order and most fibrous 
materials are ordered only along the fiber axis. 
3. Some organic crystals, when conveniently heated, assume a state 
intermediate between solid and liquid, which is called the mesomorphic 
or liquid crystal state. 
These examples indicate that periodicity can be observed to a lesser or 
greater extent in crystals, depending on their nature and on the thermo- 
dynamic conditions of their formation. It is therefore useful to introduce the 
concept of a real crystal to stress the differences from an ideal crystal with 
perfect periodicity. Although non-ideality may sometimes be a problem, 
more often it is the cause of favourable properties which are widely used in 
materials science and in solid state physics. 
In this chapter the symmetry rules determining the formation of an ideal 
crystalline state are considered (the reader will find a deeper account in 
some papers devoted to the subject, or some exhaustive or in the 
theoretical sections of the International Tables for Cryst~llography).[~~ 
In order to understand the periodic and ordered nature of crystals it is 
necessary to know the operations by which the repetition of the basic 
molecular motif is obtained. An important step is achieved by answering the 
following question: given two identical objects, placed in random positions 
and orientations, which operations should be performed to superpose one 
object onto the other? 
The well known coexistence of enantiomeric molecules demands a 
second question: given two enantiomorphous (the term enantiomeric will 
only be used for molecules) objects, which are the operations required to 
superpose the two objects? 
An exhaustive answer to the two questions is given by the theory of 
isometric transformations, the basic concepts of which are described in 
Appendix 1.A, while here only its most useful results will be considered. 
Two objects are said to be congruent if to each point of one object 
corresponds a point of the other and if the distance between two points of 
one object is equal to the distance between the corresponding points of the 
other. As a consequence, the corresponding angles will also be equal in 
absolute value. In mathematics such a correspondence is called isometric. 
The congruence may either be direct or opposite, according to whether 
the corresponding angles have the same or opposite signs. If the congruence 
is direct, one object can be brought to coincide with the other by a 
convenient movement during which it behaves as a rigid body. The 
movement may be: 
(1) a translation, when all points of the object undergo an equal 
displacement in the same direction; 
(2) a rotation around an axis; all points on the axis will not change their 
position; 
Symmetry in crystals ( 3 
(3) a rototranslation or screw movement, which may be considered as the 
combination (product) of a rotation around the axis and a transl-ation 
along the axial direction (the order of the two operations may be 
exchanged). 
If the congruence is opposite, then one object will be said to be 
enantiomorphous with respect to the other. The two objects may be brought 
to coincidence by the following operations: 
(1) a symmetry operation with respect to a point, known as inversion; 
(2) a symmetry operation with respect to a plane, known as reflection; 
(3) the product of a rotation around an axis by an inversion with respect to 
a point on the axis; the operation is called rotoinversion; 
(4) the product of a reflection by a translation parallel to the reflection 
plane; the plane is then called a glide plane. 
(5) the product of a rotation by a reflection with respect to a plane 
perpendicular to the axis; the operation is called rotoreflection. 
Symmetry elements 
Suppose that the isometric operations described in the preceding section, 
not only bring to coincidence a couple of congruent objects, but act on the 
entire space. If all the properties of the space remain unchanged after a 
given operation has been carried out, the operation will be a symmetry 
operation. Symmetry elements are points, axes, or planes with respect to 
which symmetry operations are performed. . 
In the following these elements will be considered in more detail, while 
the description of translation operators will be treated in subsequent 
sections. 
Axes of rotational symmetry 
If all the properties of the space remain unchanged after a rotation of 2nIn 
around an axis, this will be called a symmetry axis of order n ; its written 
symbol is n. We will be mainly interested (cf. p. 9) in the axes 1, 2, 3, 4, 6. 
Axis 1 is trivial, since, after a rotation of 360" around whatever direction 
the space properties will always remainthe same. The graphic symbols for 
the 2, 3, 4, 6 axes (called two-, three-, four-, sixfold axes) are shown in 
Table 1.1. In the first column of Fig. 1.1 their effects on the space are 
illustrated. In keeping with international notation, an object is represented 
by a circle, with a + or - sign next to it indicating whether it is above or 
below the page plane. There is no graphic symbol for the 1 axis. Note that a 
4 axis is at the same time a 2 axis, and a 6 axis is at the same time a 2 and a 
3 axis. 
4 1 Carmelo Giacovazzo 
Fig. 1.1. Arrangements of symmetry-equivalent 
objects as an effect of rotation, inversion, and 
screw axes. 
Table 1.1. Graphical symbols for symmetry elements: (a) axes normal to the pfane of 
projection; (b) axes 2 and 2, ,parallel to the plane of projection; (c) axes parallel or 
inclined to the plane of projection; (d) symmetry pfanes normar to the plane of 
projection; (e) symmetry planes parallel to the plane of projection 
Symmetry in crystals 1 5 
Axes of rototranslation or screw axes 
A rototranslational symmetry axis will have an order n and a translational 
component t , if all the properties of the space remain unchanged after a 
2nln rotation around the axis and the translation by t along the axis. On p. 
10 we will see that in crystals only screw axes of order 1, 2, 3, 4, 6 can exist 
with appropriate translational components. 
Axes of inversion 
An inversion axis of order n is present when all the properties of the space 
remain unchanged after performing the product of a 2nln rotation around 
the axis by an inversion with respect to a point located on the same axis. 
The written symbol is f i (read 'minus n' or 'bar n'). As we shall see on p. 9 
we will be mainly interested in 1, 2, 3, 4, 6 axes, and their graphic symbols 
are given in Table 1.1, while their effects on the space are represented in the 
second column of Fig. 1.1. According to international notation, if an object 
is represented by a circle, its enantiomorph is depicted by a circle with a 
comma inside. When the two enantiomorphous objects fall one on top of 
the other in the projection plane of the picture, they are represented by a 
single circle divided into two halves, one of which contains a comma. To 
each half the appropriate + or - sign is assigned. 
We may note that: 
(1) the direction of the i axis is irrelevant, since the operation coincides 
with an inversion with respect to a point; 
(2) the 2 axis is equivalent to a reflection plane perpendicular to it; the 
properties of the half-space on one side of the plane are identical to 
those of the other half-space after the reflection operation. The written 
symbol of this plane is m; 
(3) the 3 axis is equivalent to the product of a threefold rotation by an 
inversion: i.e. 3 = 31; 
(4) the 4 axis is also a 2 axis; 
(5) the 6 axis is equivalent to the product of a threefold rotation by a 
reflection with respect to a plane normal to it; this will be indicated by 
6 = 3/m. 
Axes of rotoreflection 
A rotoreflection axis of order n is present when all the properties of the 
space do not change after performing the product of a 2nln rotation around 
an axis by a reflection with respect to a plane normal to it. The written 
symbol of this axis is fi. The effects on the space of the 1, 2, 3 , 4, 6 axes 
coincide with those caused by an inversion axis (generally of a different 
order). In particular: i = m, 2 = 1, 3 = 6, 4 = 4, 6 = 3. From now on we will 
no longer consider the ii axes but their equivalent inversion axes. 
6 1 Carmelo Giacovazzo 
/Cl /Cl /Cl ,,Cl /Cl ,,Cl ,Cl 
o , ? ? ? ? ? ? 
H H H H H H H 
/C1 /Cl /Cl /Cl &C1 7 1 /Cl 
H H H H 
Reflection planes with translational component (glide 
planes) 
A glide plane operator is present if the properties of the half-space on one 
side of the plane are identical to those of the other half-space after the 
product of a reflection with respect to the plane by a translation parallel to 
the plane. On p. 11 we shall see which are the glide planes found in crystals. 
Symmetry operations relating objects referred by direct congruence are 
called proper (we will also refer to proper symmetry axes) while those 
relating objects referred by opposite congruence are called improper (we 
will also refer to improper axes). 
Lattices 
Translational periodicity in crystals can be conveniently studied by con- 
sidering the geometry of the repetion rather than the properties of the motif 
which is repeated. If the motif is periodically repeated at intervals a, b, and 
c along three non-coplanar directions, the repetition geometry can be fully 
described by a periodic sequence of points, separated by intervals a, b, c 
along the same three directions. This collection of points will be called a 
lattice. We will speak of line, plane, and space lattices, depending on 
whether the periodicity is observed in one direction, in a plane, or in a 
three-dimensional space. An example is illustrated in Fig. 1.2(a), where 
HOCl is a geometrical motif repeated at intervals a and b. If we replace the 
molecule with a point positioned at its centre of gravity, we obtain the 
lattice of Fig. 1.2(b). Note that, if instead of placing the lattice point at the 
centre of gravity, we locate it on the oxygen atom or on any other point of 
the motif, the lattice does not change. Therefore the position of the lattice 
with respect to the motif is completely arbitrary. 
If any lattice point is chosen as the origin of the lattice, the position of 
any other point in Fig. 1.2(b) is uniquely defined by the vector 
where u and v are positive or negative integers. The vectors a and b define a 
parallelogram which is called the unit cell: a and b are the basis vectors of 
the cell. The choice of the vectors a and b is rather arbitrary. In Fig. 1.2(b) 
four possible choices are shown; they are all characterized by the property 
that each lattice point satisfies relation (1.1) with integer u and v. 
Nevertheless we are allowed to choose different types of unit cells, such 
as those shown in Fig. 1.2(c), having double or triple area with respect to 
those selected in Fig. 1.2(b). In this case each lattice point will still satisfy 
(1.1) but u and v are no longer restricted to integer values. For instance, the 
point P is related to the origin 0 and to the basis vectors a' and b' through 
( 4 v) = (112, 112). 
The different types of unit cells are better characterized by determining 
the number of lattice points belonging to them, taking into account that the 
~ i ~ . (a) R~~~~~~~~~ of a graphical motif as an points on sides and on corners are only partially shared by the given cell. 
example of a two-dimensional crystal; (b) The cells shown in Fig. 1.2(b) contain only one lattice point, since the 
corresponding lattice with some examples Of four points at the corners of each cell belong to it for only 114. These cells primitive cells; (c) corresponding lattice with 
some examples of multiple cells. are called primitive. The cells in Fig. 1.2(c) contain either two or three 
Symmetry in crystals 1 7 
points and are called multiple or centred cells. Several kinds of multiple 
cells are possible: i.e. double cells, triple cells, etc., depending on whether 
they contain two, three, etc. lattice points. 
The above considerations can be easily extended to linear and space 
lattices. For the latter in particular, given an origin 0 and three basis 
I I 
vectors a, b, and c, each node is uniquely defined by the vector 
(1.2) = ua + ub + WC. 
The three basis vectors define a parallelepiped, called again a unit cell. a 
The directions specified by the vectors a, b, and c are the X, Y, Z Fig. Notation for a unit cell. 
crystallographic axes, respectively, while the angles between them are 
indicated by a, 0, and y, with a opposing a, opposingb, and y opposing 
c (cf. Fig. 1.3). The volume of the unit cell is given by 
where the symbol '.' indicates the scalar product and the symbol ' A ' the 
vector product. The orientation of the three crystallographic axes is usually 
chosen in such a way that an observer located along the positive direction of 
c sees a moving towards b by an anti-clockwise rotation. The faces of the 
unit cell facing a, b, and c are indicated by A, B, C, respectively. If the 
chosen cell is primitive, then the values of u, u, w in (1.2) are bound to be 
integer for all the lattice points. If the cell is multiple then u, u, w will have 
rational values. To characterize the cell we must recall that a lattice point at 
vertex belongs to it only for 1/8th, a point on a edge for 114, and one on a 
face for 112. 
The rational properties of lattices 
Since a lattice point can always be characterized by rational numbers, the 
lattice properties related to them are called rational. Directions defined by 
two lattice points will be called rational directions, and planes defined by 
three lattice points rational planes. Directions and planes of this type are 
also called crystallographic directions and planes. 
Crystallographic directions 
Since crystals are anisotropic, it is necessary to specify in a simple way 
directions (or planes) in which specific physical properties are observed. 
Two lattice points define a lattice row. In a lattice there are an infinite 
number of parallel rows (see Fig. 1.4): they are identical under lattice 
translation and in particular they have the same translation period. 
A lattice row defines a crystallographic direction. Suppose we have chosen a 
primitive unit cell. The two lattice vectors Q ,,,, and Q ,,,,,,, ,, with u, u, 
w, and n integer, define two different lattice points, but only one direction. 
This property may be used to characterize a direction in a unique way. For 
instance, the direction associated with the vector Q9,,,, can be uniquely 
defined by the vector Q,,,,, with no common factor among the indices. This 
direction will be indicated by the symbol [3 1 21, to be read as 'three, one, 
two' and not 'three hundred and twelve'. Fig. 1.4. Lattice rows and planes. 
8 1 Carmelo Giacovazzo 
When the cell is not primitive u, v, w, and n will be rational numbers. 
Thus Q112,312,-113 and Q512,1512,-5,3 define the same direction. The indices of 
the former may therefore be factorized to obtain a common denominator 
and no common factor among the numerators: Q,,,,,,, ,-,, , = Q3,6,,,6,-,,6 + 
[3 9 -21 to be read 'three', nine, minus two'. 
Crystallographic planes 
Three lattice points define a crystallographic plane. Suppose it intersects the 
three crystallographic axes X , Y , and Z at the three lattice points ( p , 0, 0 ) ) 
(0 , q , 0 ) and (0, 0 , r ) with integer p, q, r (see Fig. 1.5). Suppose that m is 
the least common multiple of p, q , r. Then the equation of the plane is 
x'lpa + y ' lqb + z ' lrc = 1. 
If we introduce the fractional coordinates x = x ' la , y = y ' lb , z = z l / c , the 
equation of the plane becomes 
x l p + y lq + z l r = 1. (1.3) 
Multiplying both sides by m we obtain 
hx + ky + lz = m (1.4) 
Fig. 1.5. Some lattice planes of the set (236). 
where h , k , and 1 are suitable integers, the largest common integer factor of 
which will be 1. 
We can therefore construct a family of planes parallel to the plane (1.4), 
by varying m over all integer numbers from -m to +m. These will also be 
crystallographic planes since each of them is bound to pass through at least 
three lattice points. 
The rational properties of all points being the same, there will be a plane 
of the family passing through each lattice point. For the same reason each 
lattice plane is identical to any other within the family through a lattice 
translation. 
Let us now show that (1.4) represents a plane at a distance from the 
origin m times the distance of the plane 
The intercepts of the plane (1.5) on X, Y , Z will be l l h , l l k and 111 
respectively and those of (1.4) m l h , m l k , ml l . It is then clear that the 
distance of plane (1.4) from the origin is m times that of plane (1.5). The 
first plane of the family intersecting the axes X , Y , and Z at three lattice 
points is that characterized by a m value equal to the least common mutiple 
of h , k , I. We can therefore conclude that eqn (1.4) defines, as m is varied, a 
family of identical and equally spaced crystallographic planes. The three 
indices h , k , and 1 define the family and are its Miller indices. To indicate 
that a family of lattice planes is defined by a sequence of three integers, 
these are included within braces: ( h k I ) . A simple interpretation of the 
three indices h, k , and I can be deduced from (1.4) and (1.5). In fact they 
indicate that the planes of the family divide a in h parts, b in k parts, and c 
in 1 parts. 
Crystallographic planes parallel to one of the three axes X, Y, or Z are 
defined by indices of type (Okl), (hol), or (hkO) respectively. Planes parallel 
to faces A, B, and C of the unit cell are of type (hOO), (OkO), and (001) 
respectively. Some examples of crystallographic planes are illustrated in Fig. 
1.6. 
As a numerical example let us consider the plane 
which can be written as 
The first plane of the family with integer intersections on the three axes will 
be the 30th (30 being the least common multiple of 10, 15, and 6) and all the 
planes of the family can be obtained from the equation lox + 15y + 62 = m, 
by varying m over all integers from -m to +m. We observe that if we divide 
p, q, and r in eqn (1.6) by their common integer factor we obtain 
x/3 + y/2 + z/5 = 1, from which 
Planes (1.7) and (1.8) belong to the same family. We conclude that a 
family of crystallographic planes is always uniquely defined by three indices 
h, k, and 1 having the largest common integer factor equal to unity. 
Symmetry restrictions due to the lattice 
periodicity and vice versa 
Suppose that the disposition of the molecules in a crystal is compatible with 
an n axis. As a consequence the disposition of lattice points must also be 
compatible with the same axis. Without losing generality, we will assume 
that n passes through the origin 0 of the lattice. Since each lattice point has 
identical rational properties, there will be an n axis passing through each 
and every lattice point, parallel to that passing through the origin. In 
particular each symmetry axis will lie along a row and will be perpendicular 
to a crystallographic plane. 
Let T be the period vector of a row passing through 0 and normal to n. 
We will then have lattice points (see Fig. 1.7(a)) at T, -T, T', and T". The 
vector T' - T" must also be a lattice vector and, being parallel to T, we will 
have T' - T" = mT where m is an integer value: in a scalar form 
2 cos (2nIn) = m (m integer). (1.9) 
Equation (1.9) is only verified for n = 1, 2, 3, 4, 6. It is noteworthy that a 
5 axis is not allowed, this being the reason why it is impossible to pave a 
room only with pentagonal tiles (see Fig. 1.7(b). 
A unit cell, and therefore a lattice, compatible with an n axis will also be 
compatible with an ii axis and vice versa. Thus axes I, 3, 3, 4, 6 will also be 
Symmetry in crystals 1 9 
0 k. (110) (010) 
(zio) 
Fig. 1.6. Miller indices for some crystallographic 
planes parallel to Z ( Z i s supposed to be normal 
to the page). 
(b) 
Fig. 1.7.(a) Lattice points in a plane normal to the 
symmetry axis n passing through 0. (b) Regular 
allowed. pentagons cannot fill planar space. 
10 1 Carmelo Giacovazzo 
Let us now consider the restrictions imposed by the periodic nature of 
crystals on the translational components t of a screw axis. Suppose that this 
lies along a row with period vectorT. Its rotational component must 
correspond to n = 1, 2, 3, 4, 6. If we apply the translational component n 
times the resulting displacement will be nt. In order to maintain the 
periodicity of the crystal we must have nt =pT, with integer p , or 
For instance, for a screw axis of order 4 the allowed translational 
components will be (0/4)T, (1/4)T, (2/4)T, (3/4)T, (4/4)T, (5/4)T, . . .; of 
these only the first four will be distinct. It follows that: 
(1) in (1.10) p can be restricted within 0 s p < n ; 
(2) the n-fold axis may be thought as a special screw with t = 0. The nature 
Fig. 1.8. Screw axes: arrangement of symmetry- 
equivalent objects. 
of a screw axis is completely defined by the symbol n,. The graphic 
symbols are shown in Table 1.1: the effects of screw axes on the 
surrounding space are represented in Fig. 1.8. Note that: 
If we draw a helicoidal trajectory joining the centres of all the objects 
related by a 3, and by a 32 axis, we will obtain, in the first case a 
right-handed helix and in the second a left-handed one (the two helices 
are enantiomorphous). The same applies to the pairs 4, and 4,, 61 and 6,, 
and 6, and 6,. 
4, is also a 2 axis, 6, is also a 2 and a 32, 64 is also a 2 and a 3,, and 63 is 
also a 3 and a 2,. 
Symmetry in crystals I 11 
We will now consider the restrictions imposed by the periodicity on the 
translation component t of a glide plane. If we apply this operation twice, 
the resulting movement must correspond to a translation equal to pT, where 
p may be any integer and T any lattice vector on the crystallographic plane 
on which the glide lies. Therefore 2t =pT , i.e. t = ( p / 2 ) T . As p varies over 
all integer values, the following translations are obtained OT, (1/2)T, 
(2/2)T, (3/2)T, . . . of which only the first two are distinct. For p = 0 the 
glide plane reduces to a mirror m. We will indicate by a, b, c axial glides 
with translational components equal to a / 2 , b / 2 , c / 2 respectively, by n the 
diagonal glides with translational components (a + b ) / 2 or (a + c ) /2 or 
( b + c ) / 2 or ( a + b + c) /2 . 
In a non-primitive cell the condition 2t = p T still holds, but now T is a 
lattice vector with rational components indicated by the symbol d. The 
graphic symbols for glide planes are given in Table 1.1. 
Point groups and symmetry classes 
In crystals more symmetry axes, both proper and improper, with or without 
translational components, may coexist. We will consider here only those 
combinations of operators which do not imply translations, i.e. the 
combinations of proper and improper axes intersecting in a point. These are 
called point groups, since the operators form a mathematical group and 
leave one point fixed. The set of crystals having the same point group is 
called crystal class and its symbol is that of the point group. Often point 
group and crystal class are used as synonyms, even if that is not correct in 
principle. The total number of crystallographic point groups (for three- 
dimensional crystals) is 32, and they were first listed by Hessel in 1830. 
The simplest combinations of symmetry operators are those characterized 
by the presence of only one axis, which can be a proper axis or an inversion 
one. Also, a proper and an inversion axis may be simultaneously present. 
The 13 independent combinations of this type are described in Table 1.2. 
When along the same axis a proper axis and an inversion axis are 
simultaneously present, the symbol n/ri is used. Classes coinciding with 
other classes already quoted in the table are enclosed in brackets. 
The problem of the coexistence of more than one axis all passing by a 
common point was first solved by Euler and is illustrated, with a different 
approach, in Appendix 1.B. Here we only give the essential results. Let us 
suppose that there are two proper axes I , and l2 intersecting in 0 (see Fig. 
1.9). The I , axis will repeat in Q an object originally in P, while 1 , will 
Table 1.2. Single-axis crystallographic point groups 
Proper axis Improper axis Proper and improper 
axis 
1 1 (i[i = i) 
2 = m- 212-= 2jm 
3 3 - 31 (3[3 = 3) 
4 4 414 = 4/m 
6 6 = 3/m 616 = 6/m 
5 + 5 + 3 = 13 
12 ( Carmelo Giacovazzo 
11 Table 1.3. For each combination of symmetry axes the minimum angles between axes t.' are given. For each angle the types of symmetry axes are quoted in parentheses Combination of cu (ded B (ded Y (ded symmetry axes 
. - 
P '. 2 2 2 90 (22) 90 (2 2) 90 (22) 
-- 
- - - 0 3 2 2 90 (2 3) 90 (2 3) 60 (2 2) 
4 2 2 90 (2 4) 90 (2 4) 45 (2 2) 
6 2 2 90 (2 6) 90 (2 6) 30 (2 2) 
0 2 3 3 54 44'08" (2 3) 54 44'08" (2 3) 70 31 '44" (3 3) 
Fig. 1.9. Arrangement of equivalent objects 4 3 2 35 15'52" (2 3) 45 (2 4) 54 44'08" (4 3) 
around two intersecting symmetry axes. 
repeat in R the object in Q. P and Q are therefore directly congruent and 
this implies the existence of another proper operator which repeats the 
object in P directly in R. The only allowed combinations are n22, 233, 432, 
532 which in crystals reduce to 222, 322, 422, 622, 233, 432. For these 
combinations the smallest angles between the axes are listed in Table 1.3, 
while their disposition in the space is shown in Fig. 1.10. Note that the 
combination 233 is also consistent with a tetrahedral symmetry and 432 with 
a cubic and octahedral symmetry. 
Suppose now that in Fig. 1.9 1, is a proper axis while 1, is an inversion 
one. Then the objects in P and in Q will be directly congruent, while the 
object in R is enantiomorphic with respect to them. Therefore the third 
operator relating R to P will be an inversion axis. We may conclude that if 
one of the three symmetry operators is an inversion axis also another must 
be an inversion one. In Table 1.4 are listed all the point groups 
characterized by combinations of type PPP, PII, IPI, IIP (P=proper, 
I = improper), while in Table 1.5 the classes with axes at the same time 
proper and improper are given. In the two tables the combinations 
coinciding with previously considered ones are closed within brackets. The 
Fig. 1.10. Arrangement of proper symmetry 
axes for six point groups. 
Symmetry in crystals 1 13 
Table 1.4. Crystallographic point groups with more than one axis 
P P P P I 1 I P I I I P 
4 3 2 (43 2 ---- 
r n l m 
Table 1.5. Crystallographic point groups with more than one axis, each axis being 
proper and improper simultaneously 
results so far described can be easily derived by recalling that: 
If two of the three axes are symmetry equivalent, they can not be one 
proper and one improper; for example, the threefold axes in 233 are 
symmetry referred by twofold axes, while binary axes in 422 differing by 
45" are not symmetry equivalent. 
If an even-order axis and a ? axis (or an m plane) coexist, there will also 
be an m plane (or a ? axis) normal to the axis and passing through the 
intersection point. Conversely, if m and ? coexist, there will also be a 2 
axis passing through ? and normal to m. 
In Tables 1.2, 1.4, and 1.5 the symbol of each point group does not reveal 
all the symmetry elements present: for instance, the complete list of 
symmetry elements in the class 2/m33 is 2/m 2/m 2/m?3333. On the other 
hand, the symbol 2/m% is too extensive, since only two symmetry operators 
are independent. In Table 1.6 are listed the conventional symbols used for 
the 32 symmetry classes. It may be noted that crystals with inversion 
symmetry operators have an equal number of 'left' and 'right' moieties; 
these parts, when considered separately, are one the enantiomorph of the 
other. 
The conclusions reached so far do not exclude the possibility of crystal- 
lizing molecules with a molecular symmetry different from that of the 32 
point groups (for instance with a 5 axis). In any case the symmetry of thecrystal will belong to one of them. To help the reader, some molecules and 
their point symmetry are shown in Fig. 1.11. 
It is very important to understand how the symmetry of the physical 
properties of a crystal relates to its point group (this subject is more 
extensively described in Chapter 9). Of basic relevance to this is a postulate 
E E 
E E E 
E E E E E k k b m m 
. - 2 E 3 4 1 m m o E E 
E E 
E E E E E E E E k E n m m 
. - 2 E 9 3 ~ ~ m w E E 
Symmetry in crystals 1 15 
2. The variation of the refractive index of the crystal with the vibration 
direction of a plane-polarized light wave is represented by the optical 
indicatrix (see p. 607). This is in general a three-axis ellipsoid: thus the 
lowest symmetry of the property 'refraction' is 2/m 2lm 2/m, the point 
group of the ellipsoid. In crystal classes belonging to tetragonal, trigonal, or 
hexagonal systems (see Table 1.6) the shape of the indicatrix is a rotational 
ellipsoid (the axis is parallel to the main symmetry axis), and in symmetry 
classes belonging to the cubic system the shape of the indicatrix is a sphere. 
For example, in the case of tourmaline, with point group 3m, the ellipsoid is 
a revolution around the threefold axis, showing a symmetry higher than that 
of the point group. 
We shall now see how it is possible to guess about the point group of a 
crystal through some of its physical properties: 
1. The morphology of a crystal tends to conform to its point group 
symmetry. From a morphological point of view, a crystal is a solid body 
bounded by plane natural surfaces, the faces. The set of symmetry- 
equivalent faces constitutes a form: the form is open if it does not enclose 
space, otherwise it is closed. A crystal form is named according to the 
number of its faces and to their nature. Thus a pedion is a single face, a 
pinacoid is a pair of parallel faces, a sphenoid is a pair of faces related by a 
diad axis, a prism a set of equivalent faces parallel to a common axis, a 
pyramid is a set of planes with equal angles of inclination to a common axis, 
etc. The morphology of different samples of the same compound can show 
different types of face, with different extensions, and different numbers of 
edges, the external form depending not only on the structure but also on the 
chemical and physical properties of the environment. For instance, galena 
crystals (PbS, point group m3m) tend to assume a cubic, cube-octahedral, 
or octahedral habit (Fig. 1.12(a)). Sodium chloride grows as cubic crystals 
from neutral aqueous solution and as octahedral from active solutions (in 
the latter case cations and anions play a different energetic role). But at the 
same temperature crystals will all have constant dihedral angles between 
corresponding faces (J. B. L. Rome' de l'Ile, 1736-1790). This property, the 
observation of which dates back to N. Steno (1669) and D. Guglielmini 
(1688), can be explained easily, following R. J. Haiiy (1743-1822), by 
considering that faces coincide with lattice planes and edges with lattice 
rows. Accordingly, Miller indices can be used as form symbols, enclosed in 
braces: {hkl). The indices of well-developed faces on natural crystals tend 
to have small values of h, k, 1, (integers greater than six are rarely 
involved). Such faces correspond to lattice planes with a high density of 
lattice points per unit area, or equivalently, with large intercepts alh, blk, 
cll on the reference axes (Bravais' law). An important extension of this law 
is obtained if space group symmetry (see p. 22) is taken into account: screw 
axes and glide planes normal to a given crystal face reduce its importance 
(Donnay-Harker principle). 
The origin within the crystal is usually chosen so that faces (hkl) and 
(h i t ) are parallel faces an opposite sides of the crystal. In Fig. 1.13 some 
idealized crystal forms are shown. 
The orientation of the faces is more important than their extension. The 
orientations can be represented by the set of unit vectors normal to them. 
This set will tend to assume the point-group symmetry of the given crystal 
(b) 
Fig. 1.12. (a) Crystals showing cubic or cube- 
octahedral or octahedral habitus, (b) crystal with 
a sixfold symmetry axis. 
16 1 Carmelo Giacovazzo 
Fig. 1.13. Some simple crystal forms: (a) 
cinnabar, HgS, class 32; (b) arsenopyritf, FeAsS, 
class mmm; (c) ilmenite, FeTiO,, class 3; (d) 
gypsum, CaSO,, class 2/m. 
independently of the morphological aspect of the samples. Thus, each 
sample of Fig. 1.12(a) shows an m3m symmetry, and the sample in Fig. 
1.12(b) shows a sixfold symmetry if the normals to the faces are considered 
instead of their extensions. The morphological analysis of a crystalline 
sample may be used to get some, although not conclusive, indication, of its 
point-group symmetry. 
2. Electrical charges of opposite signs-may appear at the two hands of a 
polar axis of a crystal subject to compression, because of the piezoelectric 
effect (see p. 619). A polar axis is a rational direction which is not symmetry 
equivalent to its opposite direction. It then follows that a polar direction can 
only exist in the 21 non-centrosymmetric point groups (the only exception is 
the 432 class, where piezoelectricity can not occur). In these groups not all 
directions are polar: in particular a direction normal to an even-order axis 
or to a mirror plane will never be polar. For instance, in quartz crystals 
(SOz, class 32), charges of opposite sign may appear at the opposite hands 
of the twofold axes, but not at those of the threefold axis. 
3. A point group is said to be polar if a polar direction, with no other 
symmetry equivalent directions, is allowed. Along this direction a per- 
manent electric dipole may be measured, which varies with temperature 
(pyroelectric effect, see p. 606). The ten polar classes are: 1, 2, m, mm2, 4, 
4mm, 6, 6mm, 3, 3m. Piezo- and pyroelectricity tests are often used to 
exclude the presence of an inversion centre. Nevertheless when these effects 
are not detectable, no definitive conclusion may be drawn. 
4. Ferroelectric crystals show a permanent dipole moment which can be 
changed by application of an electric field. Thus they can only belong to one 
of the ten polar classes. 
5. The symmetry of a crystal containing only one enantiomer of an 
optically active molecule must belong to one of the 11 point groups which 
do not contain inversion axes. 
6. Because of non-linear optical susceptibility, light waves passing 
through non-centrosymmetric crystals induce additional waves of frequency 
twice the incident frequency. This phenomenon is described by a third-rank 
tensor, as the piezoelectric tensor (see p. 608): it occurs in all non- 
centrosymmetric groups except 432, and is very efficientL7] for testing the 
absence of an inversion centre. 
7. Etch figures produced on the crystal faces by chemical attack reveal 
the face symmetry (one of the following 10 two-dimensional point groups). 
Point groups in one and two dimensions 
The derivation of the crystallographic point groups in a two-dimensional 
space is much easier than in three dimensions. In fact the reflection with 
respect to a plane is substituted by a reflection with respect to a line (the 
same letter m will also indicate this operation); and ii axes are not used. The 
total number of point groups in the plane is 10, and these are indicated by 
the symbols: 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm. 
The number of crystallographic point groups in one dimension is 2: they 
are 1 and m = (I). 
Symmetry in crystals 1 17 
The Laue classes 
In agreement with Neumann's principle, physical experiments do not 
normally reveal the true symmetry of the crystal: some of them, for example 
diffraction, show the symmetry one would obtain by adding an inversion 
centreto the symmetry elements actually present. In particular this happens 
when the measured quantities do not depend on the atomic positions, but 
rather on the interatomic vectors, which indeed form a centrosymmetric set. 
Point groups differing only by the presence of an inversion centre will not be 
differentiated by these experiments. When these groups are collected in 
classes they form the 11 Laue classes listed in Table 1.6. 
The seven crystal systems 
If the crystal periodicity is only compatible with rotation or inversion axes of 
order 1, 2, 3, 4, 6, the presence of one of these axes will impose some 
restrictions on the geometry of the lattice. It is therefore convenient to 
group together the symmetry classes with common features in such a way 
that crystals belonging to these classes can be described by unit cells of the 
same type. In turn, the cells will be chosen in the most suitable way to show 
the symmetry actually present. 
Point groups 1 and i have no symmetry axes and therefore no constraint 
axes for the unit cell; the ratios a:b:c and the angles a , P, y can assume any 
value. Classes 1 and are said to belong to the triclinic system. 
Groups 2, m, and 2/m all present a 2 axis. If we assume that this axis 
coincides with the b axis of the unit cell, a and c can be chosen on the lattice 
plane normal to b. We will then have a = y = 90" and P unrestricted and the 
ratios a:b:c also unrestricted. Crystals with symmetry 2, m, and 2/m belong 
to the monoclinic system. 
Classes 222, mm2, mmm are characterized by the presence of three 
mutually orthqgonal twofold rotation or inversion axes. If we assume these 
as reference axes, we will obtain a unit cell with angles a = P = y = 90" and 
with unrestricted a:b:c ratios. These classes belong to the orthorhombic 
system. 
For the seven groups with only one fourfold axis 
[4,4,4/m, 422,4mm, 42m, 4/mmm] the c axis is chosen as the direction of 
the fourfold axis and the a and b axes will be symmetry equivalent, on the 
lattice plane normal to c. The cell angles will be a = P = y = 90" and the 
ratios a:b:c = 1:l:c. These crystals belong to the tetragonal system. 
For the crystals with only one threefold or sixfold axis [3, 3, 32, 3m, 3m, 
6, 6, 6/m, 622, 6mm, 62m, 6/mm] the c axis is assumed along the three- or 
sixfold axis, while a and b are symmetry equivalent on the plane 
perpendicular to c. These point groups are collected together in the trigonal 
and hexagonal systems, respectively, both characterized by a unit cell with 
angles a = /3 = 90" and y = 120°, and ratios a: b :c = 1: 1:c. 
Crystals with four threefold axes [23, m3, 432, 43m, m3m] distributed as 
the diagonals of a cube can be referred to orthogonal unit axes coinciding 
with the cube edges. The presence of the threefold axes ensures that these 
directions are symmetry equivalent. The chosen unit cell will have a = P = 
y = 90' and ratios a :b:c = 1: 1: 1. This is called the cubic system. 
18 1 Carmelo Giacovazzo 
The Bravais lattices 
In the previous section to each crystal system we have associated a primitive 
cell compatible with the point groups belonging to the system. Each of these 
primitive cells defines a lattice type. There are also other types of lattices, 
based on non-primitive cells, which can not be related to the previous ones. 
In particular we will consider as different two lattice types which can not be 
described by the same unit-cell type. 
In this section we shall describe the five possible plane lattices and 
fourteen possible space lattices based both on primitive and non-primitive 
cells. These are called Bravais lattices, after Auguste Bravais who first listed 
them in 1850. 
Plane lattices 
An oblique cell (see Fig. 1.14(a)) is compatible with the presence of axes 1 
or 2 normal to the cell. This cell is primitive and has point group 2. 
If the row indicated by m in Fig. 1.14(b) is a reflection line, the cell must 
be rectangular. Note that the unit cell is primitive and compatible with the 
point groups m and 2mm. Also the lattice illustrated in Fig. 1.14(c) with 
a = b and y # 90" is compatible with m. This plane lattice has an oblique 
primitive cell. Nevertheless, each of the lattice points has a 2mm symmetry 
and therefore the lattice must be compatible with a rectangular system. This 
can be seen by choosing the rectangular centred cell defined by the unit 
vectors a' and b'. This orthogonal cell is more convenient because a simpler 
coordinate system is allowed. It is worth noting that the two lattices shown 
in Figs. 1.14(b) and 1.14(c) are of different type even though they are 
compatible with the same point groups. 
In Fig. 1.14(d) a plane lattice is represented compatible with the presence 
of a fourfold axis. The cell is primitive and compatible with the point groups 
4 and 4mm. 
In Fig. 1.14(e) a plane lattice compatible with the presence of a three- or 
a sixfold axis is shown. A unit cell with a rhombus shape and angles of 60" 
and 120" (also called hexagonal) may be chosen. A centred rectangular cell 
can also be selected, but such a cell is seldom chosen. 
(b) m;2mm (c) m;2mm 
I I i i I 
Fig. 1.14. The five plane lattices and the 
corresponding two-dimensional point groups. (d) 4;4mm 
Symmetry in crystals 1 19 
Table 1.7. The five plane lattices 
- 
Cell 
-- 
Type of cell Point group Lattice parameters 
of the net 
Oblique P 2 a, b, y 
Rectangular P, C 2mm a, b, y = 90" 
Square P 4mm a = b, y = 90" 
Hexagonal P 6mm a=b,y=120" 
The basic features of the five lattices are listed in Table 1.7 
Space lattices 
In Table 1.8 the most useful types of cells are described. Their fairly limited 
number can be explained by the following (or similar) observations: 
A cell with two centred faces must be of type F. In fact a cell which is at 
the same time A and B, must have lattice points at (0,1/2,1/2) and 
(1/2,0, 112). When these two lattice translations are applied one after 
the other they will generate a lattice point also at (1/2,1/2,0); 
A cell which is at the same time body and face centred can always be 
reduced to a conventional centred cell. For instance an I and A cell will 
have lattice points at positions (1/2,1/2,1/2) and (0,1/2,1/2): a lattice 
point at (1/2,0,0) will then also be present. The lattice can then be 
described by a new A cell with axes a ' = a/2, b ' = b, and c' = c (Fig. 
1.15). 
It is worth noting that the positions of the additional lattice points in 
Table 1.8 define the minimal translational components which will move an 
object into an equivalent one. For instance, in an A-type cell, an object at 
( x , y, z) is repeated by translation into ( x , y + m/2, z + n/2) with m and n 
integers: the shortest translation will be (0,1/2,1/2). 
Let us now examine the different types of three-dimensional lattices 
grouped in the appropriate crystal systems. 
Table 1.8. The conventional types of unit cell 
Symbol Type Positions of Number 
additional of lattice 
lattice points points 
per cell 
P primitive - 1 
I body centred (112,1/2, l r 2 ) 2 
A A-face centred (0,1/2,1/2) 2 
B B-face centred (1/2,0,1/2) 2 
C C-face centred (1/2,1/2,0) 2 
F All faces centred (112,112, O), (1/2,0,1/2) 2 
~0,112,112~ 4 
R Rhombohedrally (1/3,2/3,2/3), (2/3,1/3,1/3) 3 
centred (de 
scription with 
'hexagonal axes') 
J 
Fig. 1.15. Reduction of an I- and A-centred cell 
to an A-centred cell. 
20 ( Carmelo Giacovazzo 
Fig. 1.16. Monoclinic lattices: (a) reduction of a 
B-centred cell to a P cell; (b) reduction of an 
I-centred to an A-centred cell; (c) reduction of an 
F-centred to a C-centred cell; (d) reduction of a 
C-centred to a P non-monoclinic cell. 
Triclinic lattices 
Even though non-primitive cells can always be chosen, the absence of axes 
with order greater than one suggests the choice of a conventionalprimitive 
cell with unrestricted a, p, y angles and a:b:c ratios. In fact, any triclinic 
lattice can always be referred to such a cell. 
Monoclinic lattices 
The conventional monoclinic cell has the twofold axis parallel to b, angles 
a = y = 90", unrestricted p and a :b :c ratios. A B-centred monoclinic cell 
with unit vectors a, b, c is shown in Fig. 1.16(a). If we choose a' = a , 
b' = b, c' = (a + c) /2 a primitive cell is obtained. Since c' lies on the (a, c) 
plane, the new cell will still be monoclinic. Therefore a lattice with a B-type 
monoclinic cell can always be reduced to a lattice with a P monoclinic cell. 
An I cell with axes a, b, c is illustrated in Fig. 1.16(b). If we choose 
a' = a, b' = b, c' = a + c, the corresponding cell becomes an A monoclinic 
cell. Therefore a lattice with an I monoclinic cell may always be described 
by an A monoclinic cell. Furthermore, since the a and c axes can always be 
interchanged, an A cell can be always reduced to a C cell. 
An F cell with axes a, b, c is shown in Fig. 1.16(c). When choosing 
a' = a, b' = b, c' = (a + c) /2 a type-C monoclinic cell is obtained. There- 
fore, also, a lattice described by an F monoclinic cell can always be 
described by a C monoclinic cell. 
We will now show that there is a lattice with a C monoclinic cell which is 
not amenable to a lattice having a P monoclinic cell. In Fig. 1.16(d) a C cell 
with axes a, b, c is illustrated. A primitive cell is obtained by assuming 
a' = (a + b)/2 , b' = ( - a + b)/2 , c' = c, but this no longer shows the 
features of a monoclinic cell, since y' # 90°, a' = b' # c' , and the 2 axis lies 
along the diagonal of a face. It can then be concluded that there are two 
distinct monoclinic lattices, described by P and C cells, and not amenable 
one to the other. 
Orthorhombic lattices 
In the conventional orthorhombic cell the three proper or inversion axes are 
parallel to the unit vectors a, b, c, with angles a = /3 = y = 90" and general 
a:b:c ratios. With arguments similar to those used for monoclinic lattices, 
the reader can easily verify that there are four types of orthorhombic 
lattices, P, C, I, and F. 
Tetragonal lattices 
In the conventional tetragonal cell the fourfold axis is chosen along c with 
a = p = y = 90°, a = b, and unrestricted c value. It can be easily verified 
that because of the fourfold symmetry an A cell will always be at the same 
time a B cell and therefore an F cell. The latter is then amenable to a 
tetragonal I cell. A C cell is always amenable to another tetragonal P cell. 
Thus only two different tetragonal lattices, P and I , are found. 
Cubic lattices 
In the conventional cubic cell the four threefold axes are chosen to be 
parallel to the principal diagonals of a cube, while the unit vectors a, b, c 
are parallel to the cube edges. Because of symmetry a type-A (or B or C) 
Symmetry in crystals 1 21 
cell is also an F cell. There are three cubic lattices, P, I, and F which are not 
amenable one to the other. 
Hexagonal lattices 
In the conventional hexagonal cell the sixfold axis is parallel to c, with 
a = b, unrestricted c, a = /3 = 90") and y = 120". P is the only type of 
hexagonal Bravais lattice. 
Trigonal lattices 
As for the hexagonal cell, in the conventional trigonal cell the threefold axis 
is chosen parallel to c, with a = b, unrestricted c, a = /3 = 90°, and y = 120". 
Centred cells are easily amenable to the conventional P trigonal cell. 
Because of the presence of a threefold axis some lattices can exist which 
may be described via a P cell of rhombohedral shape, with unit vectors aR, 
bR, CR such that aR = bR = cR, aR = PR = YR, and the threefold axis along 
the UR + bR + CR direction (see Fig. 1.17). Such lattices may also be 
described by three triple hexagonal cells with basis vectors UH, bH, CH 
defined according to[61 
These hexagonal cells are said to be in obverse setting. Three further triple 
hexagonal cells, said to be in reverse setting, can be obtained by changing 
a H and bH to -aH and -bH. The hexagonal cells in obverse setting have 
centring points (see again Fig. 1.17)) at 
(O,O, O), I , I , I , (113,213,213) 
while for reverse setting centring points are at 
It is worth noting that a rhombohedral description of a hexagonal P lattice 
is always possible. Six triple rhombohedral cells with basis vectors a;, bk, 
Fig. 1.17. Rhombohedra1 lattice. The basis of the 
rhombohedral cell is labelled a,, b,, c,, the 
basis of the hexagonal centred cell is labelled 
a,, b,, c, (numerical fractions are calculated in 
terms of the c, axis). (a) Obverse setting; (b) the 
same figure as in (a) projected along c,. 
22 1 Carmelo Giacovazzo 
ck can be obtained from aH, bH, CH by choosing: 
U ~ = U H + C H , bk=b,+ cH, c k = - ( a H + b H ) + c H 
ak = -aH + CH, bk = -bH + CH, ck = a H + bH + cH 
and cyclic permutations of a;, bk, ck. Each triple rhomobohedral cell will 
have centring points at (O,0, O), (113,1/3,1/3), (213,213,213). 
In conclusion, some trigonal lattices may be described by a hexagonal P 
cell, others by a triple hexagonal cell. In the first case the nodes lying on the 
different planes normal to the threefold axis will lie exactly one on top of 
the other, in the second case lattice planes are translated one with respect to 
the other in such a way that the nth plane will superpose on the (n + 3)th 
plane (this explains why a rhombohedral lattice is not compatible with a 
sixfold axis). 
When, for crystals belonging to the hexagonal or trigonal systems, a 
hexagonal cell is chosen, then on the plane defined by a and b there will be 
a third axis equivalent to them. The family of planes (hkl) (see Fig. 1.18) 
divides the positive side of a in h parts and the positive side of b in k parts. 
If the third axis (say d) on the (a, b) plane is divided in i parts we can 
a introduce an extra index in the symbol of the family, i.e. (hkil). From the 
same figure it can be seen that the negative side of d is divided in h + k 
Fig. 1.18. Intersections of the set of parts, and then i = -(h + k). For instance (1 2 -3 5), (3 -5 2 I), 
crystallographic planes ( h k l ) with the three 
symmetry-equivalent a, b, daxes in trigonal and (-2 0 2 3) represent three plane families in the new notation. The 
hexagonal systems. four-index symbol is useful to display the symmetry, since (hkil), (kihl), and 
(ihkl) are symmetry equivalent planes. 
Also, lattice directions can be indicated by the four-index notation. 
Following pp. 7-8, a direction in the (a, b) plane is defined by a vector 
(P - 0) = ma + nb. If we introduce the third axis d in the plane, we can 
write (P - 0) = ma + nb + Od. Since a decrease (or increase) of the three 
coordinates by the same amount j does not change the point P, this may be 
represented by the coordinates: u = m - j, v = n - j, i = -j. 
If we choose j = (m + n)/3, then u = (2m - n)/3, v = (2n - m)/3, i = 
-(m + n)/3. In conclusion the direction [mnw] may be represented in the 
new notation as [uviw], with i = -(u + v). On the contrary, if a direction is 
already represented in the four-index notation [uviw], to pass to the 
three-index one, -i should be added to the first three indices in order to 
bring to zero the third index, i.e. [u - i v - i w]. 
A last remark concerns the point symmetry of a lattice. There are seven 
three-dimensional lattice point groups, they are called holohedries and are 
listed in Table 1.6 (note that 3m is the point symmetry of the rhombohedral 
lattice). In two dmensions four holohedries exist: 2, 2mm, 4mm, 6mm. 
The 14 Bravais lattices are illustrated in Fig. 1.19 by means of their 
conventional unit cells (see Appendix 1.C for a different type of cell). A 
detailed description of the metric properties of crystal lattices will be given 
in Chapter 2. 
The space groupsA crystallographic space group is the set of geometrical symmetry opera- 
tions that take a three-dimensional periodic object (say a crystal) into itself. 
Symmetry in crystals 1 23 
Triclinic 
Cubic 
Trigonal 
Hexagonal 
The total number of crystallographic space groups is 230. They were first 
derived at the end of the last century by the mathematicians Fedorov (1890) 
and Schoenflies (1891) and are listed in Table 1.9. 
In Fedorov's mathematical treatment each space group is represented by 
a set of three equations: such an approach enabled Fedorov to list all the 
space groups (he rejected, however, five space groups as impossible: Fdd2, 
Fddd, 143d, P4,32, P4132). The Schoenflies approach was most practical and 
is described briefly in the following. 
On pp. 11-16 we saw that 32 combinations of either simple rotation or 
inversion axes are compatible with the periodic nature of crystals. By 
combining the 32 point groups with the 14 Bravais lattices (i.e. P, I, F, . . .) 
one obtains only 73 (symmorphic) space groups. The others may be 
obtained by introducing a further variation: the proper or improper 
symmetry axes are replaced by screw axes of the same order and mirror 
planes by glide planes. Note, however, that when such combinations have 
more than one axis, the restriction that all symmetry elements must 
intersect in a point no longer applies (cf. Appendix l.B). As a consequence 
of the presence of symmetry elements, several symmetry-equivalent objects 
will coexist within the unit cell. We will call the smallest part of the unit cell 
which will generate the whole cell when applying to it the symmetry 
Fig. 1.19. The 14 three-dimensional Bravais 
lattices. 
24 1 Carmelo Giacovazzo 
Table 1.9. The 230 three-dimensional space groups arranged by crystal systems and 
point groups. Space groups (and enantiomorphous pairs) that are uniquely deter- 
minable from the symmetry of the diffraction pattern and from systematic absences (see 
p. 159) are shown in bold-type. Point groups without inversion centres or mirror planes 
are emphasized by boxes 
Crystal Point Space 
system group groups 
Triclinic [i3 p1 
i P 1 
Monoclinic P2, P2,, C2 
m Pm, PC, Cm, Cc 
2/m P2/m, P2,/m, C2/m, P2/c, P2,/c, C2/c 
Orthorhombic 12221 P222, P222,, P2,2,2, P2,2,2,, C222,, C222, F222, 1222, 
12,2121 
mm2 Pmm2, PmcP,, Pcc2, PmaP,, PcaS,, PncZ,, PmnZ,, Pba2, 
Pna2,, Pnn2, Cmm2, Cmc2,, Ccc2, Amm2, Abm2, Ama2, 
Aba2, Fmm2, Fdd2,lmm2, lba2, h a 2 
mmm Pmmm,Pnnn,Pccm,Pban,Pmma,Pnna,Pmna,Pcca, 
Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, 
Cmcm, Cmca, Cmmm, Cccm, Cmma, Ccca, Fmmm, 
Fddd, Immm, Ibam, Ibca, lmma 
Tetragonal p4, p41, p4,, p4,, 14, 14, 
4 P4. 14 
Cubic (231 P23, F23, 123, P2,3,_12,3 - 
~ m 3 , ~ d , Fm3, F a , lm3, ~ a 3 , I& 
PG2, Pa&?, F$32, Y.2 , 1432, P?,32, P4,32, l4?32 
P43-m, F43m, 143m, P43n, F43c, 1Gd 
m3m Pm_3m, Pn3n, PrnBn, Pn3m, Fm3m, Fm&, F b m , F&c, 
lm3m, la3d 
operations an asymmetric unit. The asymmetric unit is not usually uniquely 
defined and can be chosen with some degree of freedom. It is nevertheless 
obvious that when rotation or inversion axes are present, they must lie at 
the borders of the asymmetric unit. 
Symmetry in crystals 1 25 
According to the international (Hermann-Mauguin) notation, the space- 
group symbol consists of a letter indicating the centring type of the 
conventional cell, followed by a set of characters indicating the symmetry 
elements. Such a set is organized according to the following rules: 
1. For triclinic groups: no symmetry directions are needed. Only two space 
groups exist: PI and PI. 
2. For monoclinic groups: only one symbol is needed, giving the nature of 
the unique dyad axis (proper and/or inversion). Two settings are used: 
y-axis unique, z-axis unique. 
3. For orthorhombic groups: dyads (proper and/or of inversion) are given 
along x, y, and z axis in the order. Thus Pca2, means: primitive cell, 
glide plane of type c normal to x-axis, glide plane of type a normal to the 
y-axis, twofold screw axis along z. 
4. For tetragonal groups: first the tetrad (proper and/or of inversion) axis 
along z is specified, then the dyad (proper and/or of inversion) along x is 
given, and after that the dyad along [I101 is specified. For example, 
P4,lnbc denotes a space group with primitive cell, a 4 sub 2 screw axis 
along z to which a diagonal glide plane is perpendicular, an axial glide 
plane b normal to the x axis, an axial glide plane c normal to [110]. 
Because of the tetragonal symmetry, there is no need to specify 
symmetry along the y-axis. 
5. For trigonal and hexagonal groups: the triad or hexad (proper and/or of 
inversion) along the z-axis is first given, then the dyad (proper and/or of 
inversion) along x and after that the dyad (proper and/or of inversion) 
along [1?0] is specified. For example, P6,mc has primitive cell, a sixfold 
screw axis 6 sub 3 along z, a reflection plane normal to x and an axial 
glide plane c normal to [ ~ I o ] . 
6. For cubic groups: dyads or tetrads (proper and/or of inversion) along x , 
followed by triads (proper and/or of inversion) along [ I l l ] and dyads 
(proper and/or of inversion) along [110]. 
We note that: 
1. The combination of the Bravais lattices with symmetry elements with no 
translational components yields the 73 so-called symmorphic space 
groups. Examples are: P222, Cmm2, F23, etc. 
2. The 230 space groups include 11 enantiomorphous pairs: P3, (P3,), 
P3,12 (P3212), P3,21 (P3,21), P41 (P43), P4J2 (P4322), P4,&2 (P4&?,2), 
P6i (P65), P6, (P64), P6,22 (P6522), P6222 (P6422), P4,32 (P4,32). The 
( + ) isomer of an optically active molecule crystallizes in one of the two 
enantiomorphous space groups, the ( - ) isomer will crystallize in the 
other. 
3. Biological molecules are enantiomorphous and will then crystallize in 
space groups with no inversion centres or mirror planes; there are 65 
groups of this type (see Table 1.9). 
4. The point group to which the space group belongs is easily obtained from 
the space-group symbol by omitting the lattice symbol and by replacing 
26 1 Carmelo Giacovazzo 
the screw axes and the glide planes with their corresponding symmorphic 
symmetry elements. For instance, the space groups P4Jmmc, P4/ncc, 
14,lacd, all belong to the point group 4lmmm. 
5. The frequency of the different space groups is not uniform. Organic 
compounds tend to crystallize in the space groups that permit close 
packing of triaxial ellipsoids.[81 According to this view, rotation axes and 
reflection planes can be considered as rigid scaffolding which make more 
difficult the comfortable accommodation of molecules, while screw axes 
and glide planes, when present, make it easier because they shift the 
molecules away from each other. 
Mighell and Rodgers [9] examined 21 051 organic compounds of known 
crystal structure; 95% of them had a symmetry not higher than orthorhom- 
bic. In particular 35% belonged to the space group P2,/c, 13.3% to PI, 
12.4% to P2,2,2,, 7.6% to P2, and 6.9% to C21c. A more recent study by 
~ i l s o n , [ ' ~ ] based on a survey of the 54599 substances stored in the 
Cambridge Structural Database (in January 1987), confirmed Mighell and 
Rodgers' results and suggested a possible model to estimate the number Nsg 
of structures in each space group of a given crystal class: 
Nsg = Acc exp { -BccE21sg - Ccclmls,) 
where A,, is the total number of structures in the crystal class, [2],, is the 
number of twofold axes, [m],, the number of reflexion planes in the cell, B,, 
and Cc, are parameters characteristic of the crystal class in question. The 
same results cannot be applied to inorganic compounds, where ionic bonds 
are usually present. Indeed most of the 11 641 inorganic compounds 
considered by Mighell and Rodgers crystallizein space groups with 
orthorhombic or higher symmetry. In order of decreasing frequency we 
have: Fm3m, Fd3m, P6Jmmc, P2,/c, ~ m 3 m , ~ 3 m , C2/m, C2/c, . . . . 
The standard compilation of the plane and of the three-dimensional space 
groups is contained in volume A of the International Tables for Crystallog- 
raphy. For each space groups the Tables include (see Figs 1.20 and 1.21). 
1. At the first line: the short international (Hermann-Mauguin) and the 
Schoenflies symbols for the space groups, the point group symbol, the 
crystal system. 
2. At the second line: the sequential number of the plane or space group, 
the full international (Hermann-Mauguin) symbol, the Patterson symmetry 
(see Chapter 5, p. 327). Short and full symbols differ only for the 
monoclinic space groups and for space groups with point group mmm, 
4/mmm, 3m, 6/mmm, m3, m3m. While in the short symbols symmetry 
planes are suppressed as much as possible, in the full symbols axes and 
planes are listed for each direction. 
3. Two types of space group diagrams (as orthogonal projections along a 
cell axis) are given: one shows the position of a set of symmetrically 
equivalent points, the other illustrates the arrangement of the symmetry 
elements. Close to the graphical symbols of a symmetry plane or axis 
parallel to the projection plane the 'height' h (as a fraction of the shortest 
lattice translation normal to the projection plane) is printed. If h = 0 the 
height is omitted. Symmetry elements at h also occur at height h + 112. 
Symmetry in crystals 1 27 
4. Information is given about: setting (if necessary), origin, asymmetric 
unit, symmetry operations, symmetry generators (see Appendix l.E) 
selected to generate all symmetrical equivalent points described in block 
'Positions'. The origin of the cell for centrosymmetric space groups is 
usually chosen on an inversion centre. A second description is given if 
points of high site symmetry not coincident with the inversion centre occur. 
For example, for ~ n 3 n two descriptions are available, the first with origin at 
432, and the second with origin at 3. For non-centrosymmetric space groups 
the origin is chosen at a point of highest symmetry (e.g. the origin for ~ 4 2 c 
is chosen at 4lc) or at a point which is conveniently placed with respect to 
the symmetry elements. For example, on the screw axis in P2,, on the glide 
plane in PC, at la2, in P ~ a 2 ~ , at a point which is surrounded symmetrically 
by the three 2, axis in P2,2,2,. 
5. The block positions (called also Wyckoff positions) contains the 
general position (a set of symmetrically equivalent points, each point of 
which is left invariant only by application of an identity operation) and a list 
of special positions (a set of symmetrically equivalent points is in special 
position if each point is left invariant by at least two symmetry operations of 
the space group). The first three block columns give information about 
multiplicity (number of equivalent points per unit cell), Wyckoff letter (a 
code scheme starting with a at the bottom position and continuing upwards 
in alphabetical order), site symmetry (the group of symmetry operations 
which leaves invariant the site). The symbol adoptedr9] for describing the 
site symmetry displays the same sequence of symmetry directions as the 
space group symbol. A dot marks those directions which do not contribute 
any element to the site symmetry. To each Wyckoff position a reflection 
condition, limiting possible reflections, may be associated. The condition 
may be general (it is obeyed irrespective of which Wyckoff positions are 
occupied by atoms (see Chapter 3, p. 159) or special (it limits the 
contribution to the structure factor of the atoms located at that Wyckoff 
position). 
6. Symmetry of special projections. Three orthogonal projections for 
each space group are listed: for each of them the projection direction, the 
Hermann-Mauguin symbol of the resulting plane group, and the relation 
between the basis vectors of the plane group and the basis vectors of the 
space group, are given, together with the location of the plane group with 
respect to the unit cell of the space group. 
7. Information about maximal subgroups and minimal supergroups (see 
Appendix l .E) is given. 
In Figs. 1.20 and 1.21 descriptions of the space groups Pbcn and P4222 are 
respectively given as compiled in the International Tables for Crystallog- 
raphy. In order to obtain space group diagrams the reader should perform 
the following operations: 
1. Some or all the symmetry elements are traced as indicated in the 
space-group symbol. This is often a trivial task, but in certain cases 
special care must be taken. For example, the three twofold screw axes do 
not intersect each other in P2,2,2,, but two of them do in P2,2,2 (see 
Appendix 1. B) . 
28 1 Carmelo Giacovazzo 
P b c n 
No. 60 P 2,lb 2 / c 2 , / n 
Origin at i on I c 1 
Asymmetric unit OSxli; 0 OlzS: 
Symmetry operations 
Orthorhombic 
Patterson symmetry P m m m 
Fig. 1.20. Representation of the group Pbcn (as 
inlnternational Tables for Crystallography). 
CONTINUED 
Symmetry in crystals 1 29 
No. 60 P b c n 
Generatorsselected ( I ) ; ( 1 0 0 ) ; ( 0 1 0 ) ; t ( 0 . 0 , ) ) ; ( 2 ) ; ( 3 ) ; ( 5 ) 
Positions 
Mulliplicity. 
Wyckoff letter. 
Site Spmetry 
Coordinates Reflection conditions 
General : 
8 d 1 (1) x ( 2 ) x + ~ , J + : , z + ~ ( 3 ) f , p , i + i ( 4 ) x+:,y+i ,r Okl : k = 2 n 
( 5 ) x ' , j , ~ ( 6 ) x+:,y+;.5+: ( 7 ) x.J' ,z++ (8 ) x'+:,y+i,z h01: 1 = 2 n 
hkO: h +k = 2n 
hOO: h = 2n 
OkO: k = 2n 
001: 1 = 2 n 
Special: as above, plus 
Symmetry of special projections 
Along [001] c 2 m m Along [ 1001 p 2g m 
a l=a bl= b a l=ib b ' = c 
Origin at 0,O.z Origin at x.O.0 
Maximal non-isomorphic subgroups 
1 [ 2 ] P 2 , 2 2 , ( P 2 , 2 , 2 ) 1 ; 2 ; 3 ; 4 
[ 2 ] P 1 1 2 , / n ( P 2 , l c ) 1; 2; 5 ; 6 
[ 2 ] P 1 2 / c I ( P 2 / c ) 1 ; 3 ; 5 ; 7 
[ 2 ] P 2 , / b 1 1 ( P 2 , / c ) 1 ; 4 ; 5 ; 8 
[ 2 ] P b c 2 , ( P c a 2 , ) 1 ; 2; 7 ; 8 
[ 2 ] P b 2 n ( P n c 2 ) 1 ; 3; 6 ; 8 
[ 2 ] P 2 , c n ( P n a 2 , ) 1 ; 4 ; 6 ; 7 
IIa none 
IIb none 
Maximal isomorphic subgroups of lowest index 
IIc [ 3 ] P b c n ( a 1 = 3 a ) ; [ 3 ] P b c n ( b 1 = 3 b ) ; [ 3 ] P b c n ( c 1 = 3c ) 
h k l : h + k = 2 n 
hkl : h+k ,1=2n 
hkl : h + k , l = 2 n 
Along [010] p 2g m 
a'= bl= a 
Origin at O,y,O 
Minimal non-isomorphic supergroups 
I none 
I1 [2 ]Abma(Cmca) ; [2 ]Bbab(Ccca) ; [2 ]Cmcm;[2 ]1bam;[2 ]Pbcb(2a1= a ) ( P c c a ) ; 
[2 ]Prnca (2b1= b ) ( P b c m ) ; [ 2 ] P b m n(2c1= c ) ( P m n a ) 
30 1 Carmelo Giacovazzo 
No. 93 
Te t rag on al 
P 4 2 2 2 Patterson symmetry P 4/m m m 
Origin at 2 2 2 at 42 2 1 
Asymmetric unit O l x l i ; O I y S I ; 0 I z l $ 
Symmetry operations 
Fig. 1.21. Representation of the group P4,22 (as 
in International Tables for Crystallography). 2. Once conveniently located, the symmetry operators are applied to a 
point P in order to obtain the symmetry equivalent points P', P", . . . . If 
P', P , . . . , fall outside the unit cell, they should be moved inside by 
means of appropriate lattice translations. The first type of diagram is so 
obtained. 
3. New symmetry elements are then placed in the unit cell so producing the 
second type of diagram. . 
Some space group diagrams are collected in Fig. 1.22. Two simple crystal 
structures are shown in Figs 1.23 and 1.24: symmetry elements are also 
located for convenience. 
The plane and line groups 
There are 17 plane groups, which are listed in Table 1.10. In the symbol g 
stays for a glide plane. Any space group in projection will conform to one of 
these plane groups. There are two line groups: p l andpm. 
A periodic decoration of the plane according to the 17 plane groups is 
shown in Fig. 1.25. 
Symmetry in crystals 1 31 
CONTINUED No. 93 
Generators selected ( I ) ; t (I ,O,O); t (0, I ,0); t (O,O, I); (2); (3); (5) 
Positions 
Mulliplicily. 
Wyckofl kllcr. 
S~ le symmetry 
Coordinates Reflection conditions 
General: 
001: I =2n 
Special: as above, plus 
4 o . . 2 x,x,+ f , f , i f , x , + x , f , + Okl: I =2n 
4 n . . 2 x,x,$ f.1,: f,x,: x . f , ? Okl: I =2n 
4 1 . 2 . x,O,+ f,O.f O.x,O O,f,O hhl : I = 2n 
4 h 2 . . 1 , , , , z L f ,:,z+: +,i,T f , $ , f + f hkl : 1 =2n 
2 f 2 . 2 2 +,+,+ it+,+ hkl : I =2n 
2 e 2 . 2 2 O,O,+ O,O,+ hkl : I = 2n 
hkl : h + k + l = 2 n 
2 b 2 2 2 . f,f,O f , f , i hkl : I =2n 
Symmetry of special projections 
Along [OOI] p4mm 
a ' = a b l = b 
Origin at 0,O.z 
Along [I001 p2n1tn 
a ' = b b l = c 
Origin at x,O,O 
Maximal non-isomorphic subgroups 
I (2]P4,1 1 ( P 4 2 ) 1 ; 2 ; 3 ; 4 
[ 2 ] P 2 2 1 ( P 2 2 2 ) l ; 2 ; 5 ; 6 
[ 2 ] P 2 1 2 ( C 2 2 2 ) 1 ; 2 ; 7 ; 8 
IIa none 
IIb [ 2 ] P 4 , 2 2 ( c 1 = 2 c ) ; [ 2 ] P 4 , 2 2 ( c ' = 2 c ) ; ( 2 ] C 4 , 2 2 , ( a 1 = 2 a , b 1 = 2 b ) ( P 4 , 2 , 2 ) ; 
[ 2 ] F 4 , 2 2 ( a 1 = 2a ,b1= 2b , c1= 2 c ) ( 1 4 , 2 2 ) 
Maximal isomorphic subgroups of lowest index 
IIc [3]P4,22(c1= 3c) ; [ 2 ] C 4 , 2 2 (a1= 2a , b l = 2 b ) ( P 4 2 2 2 ) 
Along [I101 p 2 m m 
a'= + ( - a f b ) b l = c 
Origin at x,x,$ 
Minimal non-isomorphic supergroups 
I I21P4Jmmc; [2]P4J tncm; [21P42/nbc; [21P42/nnm; [31P4232 
I1 ( 2 ] 1 4 2 2 ; [ 2 ] P 4 2 2 ( 2 c 1 = c) 
32 1 Carmelo Giacovazzo 
+ + 
Fig. 1.22. Some space group diagrams. 
C 2 P 2/m 
0- 0 - 4 -0 0- -0 0- - 
On the matrix representation of symmetry 
operators 
A symmetry operation acts on the fractional coordinates x,y, z of a point P 
to obtain the coordinates (x', y', 2') of a symmetry-equivalent point P': 
The R matrix is the rotational component (proper or improper) of the 
symmetry operation. As we shall see in Chapter 2 its elements may be 0 , 
$1, -1 and its determinant is f 1. T is the matrix of the translational 
component of the operation. A list of all the rotation matrices needed to 
conventionally describe the 230 space groups are given in Appendix 1.D. 
0"; - c ------ 0 --- 0 - 0 + +O o+ +0 o+ -1 1 1- 
10 
- 
- 
- 
O+ :!I - 4 -0 0 + 0 - 0 - I T I - 0- o+ 0- 0- o+ +0 -0 
Symmetry in crystals / 33 
Fig. 1.23. A P2,2,2, crystal structure (G. Chiari, 
D. Viterbo, A. Gaetani Manfredotti, and C. 
Guastini (1975). Cryst. Struct. Commun., 4,561) 
and its symmetry elements (hydrogen atoms are 
not drawn). 
Fig. 1.24. A P2,/c crystal structure (M. Calleri, G. 
Ferraris, and D. Viterbo (1966). Acta Cryst., 20, 
73) and its symmetry elements (hydrogen atoms 
are not drawn). Glide planes are emphasized by 
the shading. 
34 1 Carmelo Giacovazzo 
Table 1.10. The 17 plane groups 
Oblique cell P I , ~2 
Rectangular cell pm, pg, an , ~ 2 m m , P ~ V , ~ 2 9 9 , c2mm 
Square cell p4, p4n-m~ p4gt1-1 
Hexagonal cell p3, p3rn1, p31n-1, p6, p6mrn 
When applying the symmetry operator C1 = (R,, TI) to a point at the end 
Fig. 1.25. A periodic decoration of the plane of a vector r, we obtain X' = CIX = RIX + TI. If we then apply to r' the 
according to the 17 crystallographic plane symmetry operator C2, we obtain 
groups (drawing by SYMPATI, a computer 
program by L. Loreto and M. Tonetti, pixel, 9, X = C2Xf = R2(RlX + TI) + T2 = R2RlX + R2Tl + T2. 
9-20; Nov 1990). 
Symmetry in crystals 1 35 
Since the symmetry operators form a mathematical group, a third symmetry 
operator must be present (see also pp. 11-12), 
C3 = C2C1= (R2R1, R2T1+ T2), (1.12) 
where R2Rl is the rotational component of C3 and (R2T, +T2) is its 
translational component. In particular the operator C2 = CC will be present 
and in general also the CJ operator. Because of (1.12) 
C1 = [RJ, (RJ-' + . . . + R + I)T]. (1.13) 
Let us now apply this result to the space group P6,. Once we have defined 
the R and T matrices corresponding to an anti-clockwise rototranslation of 
60" around z , we obtain all the six points equivalent to a point r by applying 
to it the operators CJ with j going from 1 to 6. Obviously C6= I and 
C6+j = CJ. For this reason we will say that the 6, operator is of order six 
(similarly 2 and m are of order two). 
If r is transferred to r ' by C = (R, T) there will also be an inverse operator 
C-' = (R', TI) which will bring r ' back to r. Since we must have C-'C = I, 
because of (1.12) we will also have R'R = I and R'T + T' = 0, and therefore 
C-1 = (R-1, -R-~T) (1.14) 
where R-l is the inverse matrix of R. In the P6, example, C-l = C5. When 
all the operators of the group can be generated from only one operator 
(indicated as the generator of the group) we will say that the group is 
cyclic. 
All symmetry operators of a group can be generated from at most three 
generators. For instance, the generators of the space group P6,22 are 6, and 
one twofold axis. Each of the 12 different operators of the group may be 
obtained as C',, j = 1, 2, . . . , 6, say the powers of 61, or as C2, the twofold 
axis operator, or as their product. We can then represent the symmetry 
operators of P6'22 as the product {C1){C2), where {C) indicates the set 
of distinct operators obtained as powers of C. Similarly there are two 
generators of the group P222 but three of the group ~ 4 3 m . In general all the 
operations of a space group may be represented by the product {C1) {C2) {C3). 
If only two generators are sufficient, we will set C3 = I, and if only one is 
sufficient, then C2 = C3 = I . The list of the generators of all point groups is 
given in Appendix 1 .E. 
So far we have deliberately excluded from our considerations the 
translation operations defined by the Bravais lattice type. When we take 
them into account, all the space-group operations may be written in "a very 
simple way. In fact the set of operations which will transfer a point r in a 
given cell into its equivalent points in any cell are: 
{ c ~ } {c2} {c3} (1.15) 
where T = mla + m2b + m3c is the set of lattice translations. 
The theory of symmetry groups will be outlined in Appendix 1.E. 
Appendices 
1 .A The isometric transformations 
It is convenient to consider a Cartesian basis (el, e2, e3). Any transforma- 
tion which will keep the distances unchanged will be called an isometry or 
36 1 Carmelo Giacovazzo 
an isometric mapping or a movement C. It will be a linear transformation, 
in the sense that a point P defined by the positional r x t o r r = xe, + ye2 + 
ze3 is related to a point P', with positional vector r' = xfel + y1e2 + zfe3 by 
the relation 
with the extra condition 
R R = I or R = R-'. 
R indicates the transpose of the matrix R and I is the identity matrix. 
We note that X and X' are the matrices of the components of the vectors 
r and r' respectively, while T is the matrix of the components of the 
translation vector t = Tlel + T,e2 + T3e3. 
A movement, leaving the distances unchanged, will also maintain the 
angles fixed in absolute value. Since the determinant of the product of two 
matrices is equal to the product of the two determinants, from (1.A.2) we 
have (det R)'= 1, and then det R = f 1. We will refer to direct or opposite 
movements and to direct or opposite congruence relating an object and its 
transform, depending on whether det R is +1 or -1. 
Direct movements 
Let us separate (l.A.l) into two movements: 
X 1 = X 0 + T 
Xo = RX. 
(1.A.3) adds to each position vector a fixed vector and corresponds 
therefore to a translation movement. (1.A.4) leaves the origin point 
invariant. In order to find the other points left invariant we have to set 
Xo = X and